On modified Korteweg – de Vries equation with a loaded term

• Date: 2021
• Main Authors: Khasanov , A. B., Allanazarova , T. Zh., Hasanov, Aknazar, Хасанов, А. Б., Алланазарова, Т. Ж.
• Format: Article
• Language: Ukrainian
• Published: Institute of Mathematics, NAS of Ukraine 2021
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/6073
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.957 In this paper, the method of the inverse spectral problem is applied to finding a solution to the Cauchy problem for the modified Korteweg–de Vries equation (mKdV) in the class of periodic infinite-gap functions. A simple derivation of the Dubrovin system of differential equations is proposed. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly converging functional series constructed from the solutions of the infinite system of Dubrovin equations and the formulas for the first trace do indeed satisfy the mKdV equation. Moreover, it was proved that: 1) if the initial function is a real $ \pi $-periodic analytic function, then the solution of the Cauchy problem for the mKdV equation with a loaded term is also a real analytic function with respect to the variable $x;$ 2) if the number $ \dfrac {\pi} {2} $ is the period (antiperiod) of the original function, then $ \dfrac {\pi} {2} $ is also the period (antiperiod) in the variable $x$ of the solution to the Cauchy problem for the mKdV equation with a loaded term.